Prove: Galois Extensions Homework - E is Isomorphic to B_N x B_H

[dabien]

Homework Statement

Prove
Suppose that B_1, B_2 are Galois extensions of F with respective Galois groups G_1, G_2, and that E=B_1 \otimes B_2 is a field. Then it is Galois over F with Galois group isomorphic to G_1 \times G_2.

#2
Suppose that E is a Galois extension of F with Galois group G and that G contains subgroups H and N with N normal in G, H \cap N={1} and HN=G. Let B_N be the fixed field of N (so B_N is Galois over F) and B_H be the fixed field of H. Prove that E is isomorphic to B_N \otimes B_H. (If H is also normal in G then G\cong H \times N giving a converse to the preceding)

Homework Equations

The Attempt at a Solution

I am trying to get started...

 

[dabien]

My progress:
For #1:
I note that one equivalent condition for an extension to be Galois is that the number of automorphisms which fix the base field equals the degree of the extension. It is clear that G_1 \times G_2 is contained in the group of automorphisms of B_1 \otimes B_2 which fix F. Now I need help trying to argue that any such automorphism is necessarily in G_1 \times G_2. I also need help trying to check orders to confirm that the extension is Galois.

For 2) I have read a condition for when the tensor product of two fields is actually equal to the product of the fields (when the fields are linearly disjoint). I need help verifying that this holds here. I also need to show that this product has to be all of E.
I have a stated lemma: If H,K are two normal subgroups of a group G such that they intersect trivially and HK=G, then G is isomorphic to H times K. I need help to show that if E=B_N \otimes B_H with H and K satisfying the above, then G=H \times K.